Question

Of the three lines x+root3y=0 ,x+y=1 and x-root3y=0 , two are equations of two altitudes of an equilateral triangle .prove that the centroid of the triangle is the point (0,0)

Answer 1

$$\begin{gathered} \mathrm{Suppose}\mathrm{the}\mathrm{equations}\mathrm{of}\mathrm{altitudes}\mathrm{of}\mathrm{the}\mathrm{equilateral}∆\mathrm{are}: \\ \mathrm{x}-\sqrt{3}\mathrm{y}=0......\left(1\right)\mathrm{and}\mathrm{x}+\sqrt{3}\mathrm{y}=0.......\left(2\right) \\ \mathrm{Now},\mathrm{from}\left(1\right),\mathrm{we}\mathrm{get}\mathrm{x}=\sqrt{3}\mathrm{y}. \\ \mathrm{Putting}\mathrm{x}=\sqrt{3}\mathrm{y}\mathrm{in}\left(2\right),\mathrm{we}\mathrm{get} \\ \sqrt{3}\mathrm{y}+\sqrt{3}\mathrm{y}=0 \\ \Rightarrow 2\sqrt{3}\mathrm{y}=0 \\ \Rightarrow \mathrm{y}=0 \\ \mathrm{Now},\mathrm{from}\left(1\right),\mathrm{we}\mathrm{get}\mathrm{x}=0 \\ \mathrm{So},\mathrm{both}\mathrm{the}\mathrm{altitudes}\mathrm{intersect}\mathrm{at}\left(0,0\right) \\ \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{altitudes}\mathrm{of}\mathrm{an}\mathrm{equilateral}∆\mathrm{intersect}\mathrm{at}\mathrm{centroid},\mathrm{so} \\ \mathrm{coordinates}\mathrm{of}\mathrm{centroid}\mathrm{are}\left(0,0\right). \end{gathered}$$